Snap to face doesn't work
First you need to adjust the rotation of the small object to be fitted on the surface with the correct angle as shown below:





After that, switch to edit mode of the big object and switch to Face selection and select the face as shown below:





Then hit Shift+S and select Cursor to Selected as shown below:





After that select the small object and hit again Shift+S and select Selected to Cursor this time and it will work as shown below:


 


Updated .blend file can be found here: